Friday, January 04, 2013
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Off-topic paragraph: Today, we celebrate 370 years from the birth of Isaac Newton, arguably the brightest scientist ever. He was born on January 4th, 1643 (New System: it's December 25th, 1642, in the Old System). He was the founder of classical physics (and, in fact, physics in the modern sense), the universal law of gravitation, co-inventor of calculus, the discoverer of lots of mathematical methods, laws in optics, and so on. He was also a reliable executioner of counterfeiters and a devout Christian whose literal belief in the Bible and in the existence of the Holy Spirit permeating the whole space actually powered his physics research.

But in the text below, we're going to discuss an example illustrating the special theory of relativity, one of the theoretical frameworks that superseded Newton's theories.

Our cars and trains and airplanes are fast but the speed is negligible relatively to the speed of light \(c\) which is why Albert Einstein's special theory of relativity remains abstract for most of us. It hasn't been hardwired in our brains.

However, all the relativistic effects are mundane at the particle accelerators such as the LHC. Protons are accelerated to speeds that are very close to the speed of light.

If you did it with a slightly higher number of protons, you could accelerate whole human beings to such speeds – assuming you would find out how to accelerate electrons as well and add them to the atoms again (it's hard to accelerate the electrically neutral human bodies directly). That's why the experience of the protons isn't something "totally different" from what humans could experience. With a little bit of extra work, we could experience it.

But what do the protons experience?

First, let us determine the speed of the protons. The rest mass of a proton is\[

m_0 = 0.938272\GeV/c^2.

\] It's almost one gigaelectronvolt (divided by the squared speed of light). However, the total energy carried by the proton is enhanced by the Lorentz gamma factor:\[

\] There are seven digits "nine" followed by a "seven". If you multiply this number "almost equal to one" by the speed of light, \(c=299,792,458\,{\rm m/s}\) (exactly), you will find out that the proton's speed is just nine meters per second smaller than the speed of light in the vacuum! The proton's speed only differs from the speed of light by the speed of an Olympic runner; however, don't forget that the simple addition of speeds isn't the right way to calculate the relative velocities in relativity.

Now, the proton's speed is associated with a particular world line in the spacetime. There is a certain kind of a "hyperbolic angle" called the rapidity \(\varphi\) in between the moving LHC proton's world line and a static proton's world line. Its value is\[

\] The angle is slightly above nine "hyperbolic/imaginary radians". Note that the rapidity isn't periodic – because the hyperbola, unlike the circle, isn't closed. Nevertheless, the total amount of acceleration that the proton had to experience from its own viewpoint is analogous to the rotation by nine radians – except that we are talking about a rotation in the Minkowski spacetime. The value of the rapidity isn't terribly high but it corresponds to speeds that are very close to the speed of light and whose \(\gamma\) is huge.

The fact that the proton has this huge speed has consequences. First of all, in the proton's instantaneous inertial reference frame, the circular LHC ring looks like an ellipse. It is a hugely squeezed ellipse, almost a line interval. This ellipse is \(4,263\) times wider than tall. It's insane. Even if your horizontal display resolution were \(4,263\) pixels, the vertical height of the picture of the LHC would be as small as one pixel. I won't even try to insert a real picture here. A horizontal line may be more appropriate.

And this is how the world "is" according to the proton's natural reference frame. Note that the proton is moving vertically right now – it is either on the left or the right endpoint of this "ellipse pretending to be a line interval". This very unusual shape of the LHC ring is no sleight-of-hand. It's how Nature works.

I have already mentioned that the total mass – or the total energy – of the LHC proton is \(4,263\) times greater than what it is when the proton is at rest. Aside from the total mass/energy, other things get expanded or shrunk by the same factor. For example, imagine that the proton is replaced by an unstable particle (such as a pion) that lives for time \(t_0\), if measured in its rest frame. For the sake of simplicity, imagine that the total lifetime is exactly \(t_0\) instead of being statistically distributed with the right average. Let's also assume that the particle flies along a straight path for a while, instead of the circular ring.

How far can such a particle get during its lifetime? In Newton's theory, you would simply say that at the given speed \(v\), very close to the speed of light \(c\), the distance traveled would be\[

s \neq vt_0.

\] I wrote \(\neq\) because Newton's theory isn't the right theory of space and time. Instead, in the lab frame, the particle may traverse a much longer distance\[

s = \gamma vt_0.

\] It is \(\gamma\approx 4,263\) times longer than it is according to Newton's theory! There are two simple ways to explain the origin of this extra factor of \(\gamma\). In the lab frame associated with the LHC physicists, all the processes occurring "inside" the moving particle are slowed down due to the "time dilation" by the factor of \(\gamma\). That's true for the "aging process" of the particle, too. Because the particle ages \(4,263\) times more slowly, it is able to fly a \(4,263\) times longer a distance during its lifetime.

The explanation of the "enhancement" is different in the particle's own inertial system but the result is the same. According to the particle's own reference frame, the longitudinal distances (in the direction of motion) are shrunk \(\gamma\approx 4,263\) times (recall the thin ellipse above). The time goes "normally" and the particle sees its lifetime as \(t_0\). So in its rest frame, it really travels over the "Newtonian" distance \(vt_0\). However, when you ultimately want to translate this distance to some actual places in France and Switzerland, you must appreciate that the particle sees shorter longitudinal (=in the direction of motion) distances due to the "Lorentz contraction", so the actual distances "on the map" are \(\gamma\approx 4,263\) times longer than they seem to be from the particle's viewpoint. Again, the distance on the map of Europe traversed over the particle's lifetime is equal to \(\gamma v t_0\).

I have discussed the mass increase, the time dilation, and the Lorentz contraction. There are lots of other unfamiliar effects at those speeds. The relativity of the simultaneity is a characteristic feature of special relativity but I won't spend much time with it in this blog entry. But there are others. For example, the proton (let's return to the proton) sees modified colors due to the Doppler effect. It's the apparent change in the frequency of a wave caused by relative motion between the source of the wave and the observer. (Reference: Sheldon. No other sitcom has ever squeezed so many valid definitions of physical concepts into a few minutes. One could argue that even most of the "popular scientific" programs contain a smaller amount of truly correct and accurate statements about physics than TBBT. Incidentally, Sheldon got accused of sexual harassment last night.)

Now, in Newton's physics, the frequency gets modified either by the factor of \(1\pm v/c\) or by the factor of \(1/(1\pm v/c)\), depending on whether the source or the observer is moving and depending on whether the motion is "away from each other" or "towards each other". In Newton's theory, it matters whether the source or the observer is moving because there's a static environment (air for sound; the notorious luminiferous aether for the light) in between which has its own preferred frame.

In special relativity, there's no luminiferous aether. There's no preferred frame. Consequently, it doesn't matter whether the source is moving, or whether the observer is moving. There is a unified formula for the Doppler change in the frequency:\[

\frac{f}{f_0} = \sqrt{ \frac{1\pm v/c}{1\mp v/c} } .

\] Note that the sign in the denominator is the opposite sign than the sign in the numerator. And if you change the sign of \(v\), it has the same effect as inverting the square root (\( f/f_0\to f_0/ f \)). Fine. So how many times do the frequencies change for our speed \(0.9999999724c\)? A simple calculation shows that the result is actually \[

\frac{f}{f_0} \approx 8,513.

\] It is no coincidence that the numerical value is approximately equal to \(2\gamma\), two times our "four thousand". That's the right approximation of the Doppler ratio for any ultrarelativistic speed (=very close to the speed of light). It means that if there is some light moving against or towards the proton, the proton observes the frequency as \(8,500\) times higher than what it is in the lab frame! If some photons are "catching up with the photon" from its back, they ultimately catch up and are "seen" by the proton but the proton sees the frequency as \(8,500\) times lower than it is in the lab frame!

So you may send two photons of the same color (according to the LHC lab frame) from two directions and the proton will think that the frequencies of the photons are different. The photon coming from the front side has \(8,513^2\approx 72,500,000\) times higher frequency than the photon coming from the opposite side. The frequency ratio observed by the proton exceeds the stunning factor of seventy-two million. And this is no science-fiction. We are talking about genuine protons at their speed they have had in the LHC. In early 2015 when the LHC restarts, the energy will be almost doubled to \(13\TeV\) – i.e. \(6.5\TeV\) per proton – so the numbers will be even more extreme: \(\gamma\approx 6,928\), \(v/c\approx 0.99999999\), \(f/f_0\approx 13,800\).

I can't resist to mention an additional extreme number, one related to collisions. In the lab frame, we had two colliding \(4\TeV\) protons. It's interesting to ask what the oppositely moving proton looks like in our proton's reference frame. The mutual speed is calculated through the relativistic formula\[

\] Thanks to "relativity" for a correction of the numerical result by a dozen of percent (the inaccurate result was due to rounding of some sort.) In the numerical form of \(V/c\), there are fifteen digits "nine" followed by a "six". The corresponding \(\gamma_{\rm relative}\) exceeds 36 million (this is the factor appearing in the Lorentz contraction and time dilation, too). So the left-moving proton sees its right-moving friend (or foe) as having a 36 million times greater energy relatively to the rest mass/energy. It means that the other proton's energy is over \(31,000\TeV\) in the first proton's inertial system. However, one can't probe the energy scale at thousands of \(\TeV\)s (which would be great for the discovery of all the superpartners) because only the total energy in the center-of-mass frame determines how deeply the LHC sees into the matter – and it's been "just" \(8\TeV\).

People aren't really observing the world through the fast protons' eyes. They always see what happens from the inertial frame of the LHC detectors. However, this single frame is enough to tell us how the world actually works. And the answer is that it works in agreement with relativity. Because relativity implies that it is legitimate to switch to any other inertial frame, including the inertial system of the fast proton, the extreme perceptions of the proton that we have discussed are genuine according to the experimental evidence that the LHC is giving us.

The special theory of relativity is a mundane set of facts and the LHC is showing the special theory of relativity every day, in every collision. It just works. One could argue that people who work with the observations at the LHC have hardwired special relativity into their brains. Most of them have gotten used to Nature's being fundamentally quantum mechanical, too.

Sometime in the distant future, you may imagine an advanced civilization that will be able to push spaceships to the same speed as the speed of the protons at the LHC. Well, I actually doubt this will ever occur but what do I know. If it occurs, you may apply the ideas above to interstellar and intergalactic space travel. Due to the time dilation and/or Lorentz contraction, an astronaut will actually be able to get "almost arbitrarily far", even thousands or millions of light years away, during his lifetime.

Dear raV, this type of warp drive traffic is forbidden by the laws of relativity although some people like to delude themselves into believing that relativity can't go this far. It can. Even in a curved space, the laws of special relativity have many consequences of various types. First, they hold for local regions of the spacetime that resemble the flat spacetime in special relativity. Second, they also hold for the long-distance structure of the spacetime where the local curvature may be neglected because it's localized. What do I mean?

From the viewpoint of infinity, the whole spaceship including the gadget to warp the space around it is still a perturbation moving in the Minkowski space, and it simply cannot move faster than light because it would violate the Lorentz invariance of the effective theory valid for the distance scales much longer than the warp drive spaceship.

Locally or microscopically, one may also see why the gadget is impossible because the required "exotic matter" that you refer to violates the (positive) energy conditions. The hypothetical engine is equivalent to something that allows the energy to go negative which is really what allows the tachyon-like (faster-than-light) behavior of the local excitations. Such things can't occur in this Universe or any other relativistic world.

If two protons are approaching each other, each at almost the speed of light, it would seem that in the reference frame of one the other is exceeding the speed of light which is not possible. Is the answer to this time dilation?

Dear Harlow, the relative speed never exceeds the speed of light. I have written the right formula to add velocities v1, v2. It's, once again.

(v1+v2) / (1+v1*v2/c^2)

You may check it is never greater than "c" if both v1,v2 are less than "c". This addition is derived from relativity and is therefore related to time dilation and other things but the formula is not "just" about time dilation.

So a photon sees nothing, experiences nothing, remembers nothing, because it knows no time? It may traverse galaxies, span the vasty deep, overfly imperial civilzations, outrun an exploding supernova, but know nothing of any of these wonders. That's no way to live :(

2) Twin A hops on a rocket and goes close to the speed of light towards a start 10 lightyears away and then comes back in a similar fashion. A is biologically younger than B.

So to explain this paradox (which it really I guess),

3) From the frame of reference of A, the distance to the star is contracted while she fells time pass normally on board the ship (A locking at clock A). Thus, she would cover that distance quickly due to the shorter distance as seen from A.

4) From the frame of reference of B, the clock A (which would be weird due to the Doppler effect) on board the ship would tick slowly and thus, according to B, twin A would transverse the full 100 ly in regular time (according to clock B) but the aging difference would be explained because of the slower ticking clock A onboard the ship.

Now, this is all good but I have an issue understanding something. It has to be symmetric right? Because from the point of being of twin A, it could be the earth moving away at close to the speed of light. So why is twin A younger than B?

The situation is not symmetric. The frame of A, the babe on Earth, is inertial and the laws of special relativity apply to it. The frame of B isn't inertial because she's been feeling acceleration, so the laws of special relativity don't apply. At the end, the traveling twin will be younger.

Alternatively, you may describe the situation in general relativity which allows *all* frames and coordinate systems. Conceptually speaking, the situation is symmetric - both frames are equally good. But the detailed numbers aren't the same in both frames. The flying, accelerating twin who experiences acceleration during the trip may be said to experience a gravitational field - gravitational field and fictitious forces such as centrifugal/inertial ones are equivalent, a fact general relativity intensely exploits. But gravitational fields slow down time, too. That's why the traveling twin will end up younger according to general relativity.

Actually this problem is solved in every SR book, because whether the travelers are accelerated or not they are still in Minkowski space and it's still flat. Einstein was very empathetic on this point. GR is only needed when ST is curved. Of course you can do the problem in GR - it works in flat ST too - but that's overkill. The trajectories are not symmetric because there is no LT that takes one into the other.